Here we show that the value of \(g_\phi (A)\)  in terms of \(f(A)\)  for any set \(A\) . In the analysis of [1]}, this follows from submodularity of \(g\) , which is inherited from the submodularity of \(f\) . Here, we must again adopt a different approach. We begin by proving the following claim. Fix some set \(A \subseteq X\)  and for all \(0 \le j \le |A|\)  define
\(F_j = \sum _{B \in \binom{A}{j}} f(B)\)  as the total value of all subsets of \(A\)  of size \(j\) . Note that since we suppose \(f\)  is normalized, \(F_0 = f(\emptyset ) = 0\) .
